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DOI: 10.1039/C6RP00170J (Paper) Chem. Educ. Res. Pract., 2016, 17, 1190-1202

How well does A-level Mathematics prepare students for the mathematical demands of chemistry degrees?

Ellie Darlington * and Jessica Bowyer
Cambridge Assessment, Cambridge CB2 1GG, UK. E-mail: darlington.e@cambridgeassessment.org.uk

Received 3rd August 2016 , Accepted 14th September 2016

First published on 16th September 2016

Abstract

332 undergraduate chemistry students were surveyed in order to establish whether they had found A-level Mathematics and/or Further Mathematics to be good preparation for their degree. Perceptions of both subjects were found to be positive, with more than 80% of participants describing Mathematics or Further Mathematics as good preparation. In particular, pure mathematics and mechanics topics were found to be the most useful. Additionally, over 90% of participants who had studied at least AS-level Further Mathematics reported that there was an overlap between the material they encountered at A-level and in the first year of undergraduate study. This indicates that prospective undergraduate chemists would significantly benefit from studying A-level Mathematics, and in particular may benefit from specialising in the study of mechanics, something which will only be possible through the study of Further Mathematics after qualifications are reformed in September 2017. Universities should consequently consider revising their entry requirements or recommendations to applicants.

1. Undergraduate chemistry in the United Kingdom

The number of students progressing to undergraduate chemistry has come under increasing scrutiny over recent years, due to the discipline being listed as a strategically important and vulnerable subject (SIVS) by the Higher Education Funding Council for England (HEFCE). Chemistry's designation as a SIVS is primarily due to concern about the number of undergraduates, the subsequent closure of university chemistry departments, and the perceived importance of the chemistry research and industrial sectors to the United Kingdom (UK) economy. In response to this, certain initiatives have been set up. For example, ‘Chemistry for Our Future’ was a £3.6 million programme funded by HEFCE in partnership with universities, schools and industry to promote chemistry-related professions and increase participation in post-compulsory chemistry. In its first year of implementation, the National Foundation for Educational Research (NFER) found that the programme had positive impacts on secondary school students' chemistry knowledge and skills, enjoyment and perceptions of the subject. At the university level, the activities provided by the programme resulted in improved performance, drop-out rates and subject-related motivation amongst chemistry undergraduates (Lord et al., 2008).

Indeed, recent data indicates that the sector is becoming healthier. The number of full-time students studying undergraduate chemistry increased by 57% between 2002 and 2014 (Higher Education Statistics Agency, 2016). In 2014/15, there were 18[thin space (1/6-em)]190 students registered on undergraduate chemistry degrees in the UK (see Fig. 1), comprising 1.4% of the UK's total full-time undergraduate population. This was an all-time high for the subject, compared to a low of 0.9% in 2005/06 when there were only 11[thin space (1/6-em)]180 chemistry undergraduates. Similarly, Advanced ‘A’ level Chemistry (see Section 2.1.1 about A-levels) is more popular than it has ever been: in 2016, chemistry candidates comprised 6.2% of all A-levels (Joint Council for Qualifications, 2016).


image file: c6rp00170j-f1.tif
Fig. 1 Full-time students of chemistry degrees, 1996–2015. Note: these figures include students of joint honours degrees with chemistry, e.g. chemistry and medicinal chemistry. Data from the Higher Education Statistics Agency (1998–2016).

In the United Kingdom, secondary school students apply to university in their final year. They choose which degree they want to study and apply to that particular course. Many universities offer ‘joint honours’ subjects where students can apply to study two subjects, e.g. history and French. However, few universities offer the opportunity to specialise in a different subject once they have begun their studies. If a student is accepted onto a chemistry degree course, it is usually the case that all of the modules that they will study as part of that degree will be taught in their chemistry department by chemistry specialists. Most modules will be compulsory in the first year, and students are often given an element of choice after that, within the bounds of their degree specialism. The lectures that chemistry undergraduates attend will usually be specifically for chemistry undergraduates, including topics in mathematics and physics which are relevant to their degree.

2. The mathematical component of undergraduate chemistry

2.1 Post-compulsory mathematics qualifications

2.1.1 A-levels. Advanced ‘A’ levels in England, Northern Ireland and Wales are qualifications taken by upper secondary school students at age 18. Study begins after students have earned their General Certificate of Secondary Education (GCSE) qualifications around age 16. Students typically study for three or four A-level subjects from a wide range of options, with study taking place over the course of two years. There are no compulsory subjects. Assessment is modular, with examinations taken at the end of the course. It is possible for students to be awarded an Advanced Subsidiary ‘AS’ level by studying just the first half of the A-level course and taking examinations at the end of the first year.

Passes in A-levels are usually required for students to be given conditional offers to study at university. A-levels are graded between A and E, with those who do not achieve an E grade being ‘ungraded’ and failing. The grades and subjects required for university entry depend on the degree subject to be studied and the university.

2.1.2 A-level Mathematics and Further Mathematics. Subject to achieving a good pass grade in GCSE Mathematics, the study of mathematics post-16 is not compulsory. However, Mathematics was the most popular A-level subject in 2016, accounting for 11% of all A-levels taken (Joint Council for Qualifications, 2016). A-level Further Mathematics, which may be studied in addition to A-level Mathematics, has been rapidly growing in popularity over recent years. In 2005, it had 5933 candidates (accounting for 0.8% of all A-levels) whereas in 2016 there were 15[thin space (1/6-em)]257 candidates (Joint Council for Qualifications, 2016). Undergraduate students of science, mathematics and social science subjects have been found to perceive it as being useful preparation for the mathematical components of their degree courses (Darlington and Bowyer, 2016a).

A-level Mathematics comprises four compulsory Core Pure Mathematics units of equal weighting, along with two applied mathematics units. The applied units may be selected from one of three strands: (1) Statistics; (2) Mechanics; and (3) Decision Mathematics. There are between two and five sequential units within each of these, depending on the strand and the awarding body. The more advanced applied mathematics units (e.g. Mechanics 3 and above) can only be studied in Further Mathematics.

Core Pure Mathematics contains topics such as proof, algebra and functions, coordinate geometry, trigonometry, sequences and series, exponentials and logarithms, calculus and vectors. Topics taught in earlier Statistics units include statistical sampling, data presentation and interpretation, probability and statistical distributions. The types of topics taught in earlier Mechanics units are often taught in other countries as part of physics qualifications (Munro, 2015), and include topics such as kinematics, forces, Newton's laws and moments. Decision Mathematics includes topics such as graphs and networks, algorithms, linear programming and game theory.

Students may take two applied units from the same strand (e.g. Statistics 1 + Statistics 2) or one from two different strands (e.g. Statistics 1 + Mechanics 1). Hence, there are six possible routes through A-level Mathematics (see Fig. 2). At AS-level, students take two compulsory Core Pure Mathematics units and one of Mechanics 1, Statistics 1 or Decision Mathematics 1. Restrictions on resources and timetabling within schools may mean that some students are given a restricted choice on what units they may study, if they do have any say.


image file: c6rp00170j-f2.tif
Fig. 2 Different routes through A-level Mathematics.

A-level Further Mathematics comprises two compulsory Further Pure Mathematics units, plus four optional units (at AS-level, Further Pure Mathematics 1 and two optional units). The optional units can be selected from any of the three applied strands or from additional Further Pure Mathematics units. There are therefore a large number of different routes through Further Mathematics.

2.1.3 Reform. A substantial reform programme is underway in England which will see A-levels become linear and the content of many subjects change. Students beginning studying A-level Mathematics from September 2017 will study an entirely different syllabus. After consultation with the A-level Content Advisory Board (ALCAB, 2014) – a panel of subject-specific experts from secondary, further and higher education – the UK government decreed that the optionality within A-level Mathematics should be removed in order that all students study exactly the same content. This will see students all studying a large proportion of pure mathematics content, as well as mechanics and statistics (Department for Education, 2014b). The aim of this change is to reduce the variability in students' mathematical backgrounds when they begin university study as the current A-level structure means that, for example, some students may arrive with no mechanics knowledge and others with very in-depth mechanics knowledge. This will make it easier for universities to design first-year courses if A-level Mathematics is an entry requirement.

Decision mathematics content has been removed from A-level Mathematics, but will be available for study as part of Further Mathematics. Further Mathematics will be entirely redeveloped, and will continue to have some compulsory pure mathematics content, but the remaining units do not have to follow the current Statistics/Mechanics/Decision Mathematics structure.

In addition to the changes to A-level Mathematics and Further Mathematics, new post-compulsory ‘Core Mathematics’ qualifications are being introduced. These will be aimed at students who achieve at least a C grade in GCSE Mathematics and who wish to continue studying some mathematics, but not to A-level. Core Mathematics will cover a range of topics in pure and applied mathematics, with a focus on real-world applications and problem solving (Department for Education, 2015). A wide range of topics are taught as part of the qualifications, including anything from estimation and algebra to the Normal distribution and probability, to financial mathematics (OCR, 2014). It is hoped that the introduction of Core Mathematics will result in a greater proportion of students studying mathematics post-16.

2.2 Mathematics entry requirements for chemistry degrees in the UK

Few universities include either A-level Mathematics or Further Mathematics as an entry requirement for their undergraduate chemistry courses. However, the majority of the highest-performing institutions require an A or B grade in A-level Mathematics, with some listing it as a ‘preferred’ rather than compulsory subject. Nonetheless, in 2010, 64% of first year chemistry undergraduates had taken A-level Mathematics (Hillman, 2014, p. 22). Table 1 shows the universities which require students to have achieved more than a C grade in GCSE Mathematics in order to be offered a place to study a three-year undergraduate chemistry degree.
Table 1 Mathematics entry requirements for undergraduate chemistrya
University Mathematics requirement and grade
GCSE A-level
a Entry requirements as of 5th September 2016 according to the University and Colleges Admissions Service (UCAS) and individual universities' websites. A total of 52 institutions were found to offer single honours three-year undergraduate chemistry degrees (‘F100’ according to UCAS definitions) in the UK for first teaching in September 2017. b Though there is only a GCSE Mathematics requirement at these universities, students are either recommended to study A-level Mathematics or must study A-level Mathematics or another science A-level in addition to A-level Chemistry in order to be accepted onto the course.
University of Bath Bb
University of Birmingham B
University of Bristol A
Durham University A
University of East Anglia Bb
University of Edinburgh A
Heriot-Watt University B
Imperial College, London A
King's College, London Ab
Lancaster University B
University of Leicester A
Newcastle University B
University of Oxford A
Reading University B
University of Sheffield Bb
University of Southampton Ab
University of St Andrews Bb
University of Sussex B
University College, London B
University of Warwick B

Gadd (2000) argues that low mathematics entry requirements have two negative effects on the ability and performance of chemistry undergraduates. Firstly, low entry requirements may be misleading as they create the impression that mathematics is not essential (or even useful) to the study of chemistry at degree level. This may consequently encourage students to apply for chemistry courses who would not have done so, had they been aware of the mathematical content. Secondly, students without A-level Mathematics have been found to struggle during their undergraduate studies, due to a lack of prior mathematical skills and knowledge. Porkess (2008) agrees, arguing that whilst some departments may be reluctant to impose higher entry requirements in fear of putting potential applicants off, failing to clarify the importance of A-level Mathematics is “sending quite the wrong signals into schools… [and is] an invitation to students to arrive at university having done inappropriate A level subjects” (p. 32).

This viewpoint is reflected in chemistry undergraduates' own accounts of their experiences of the mathematics involved at degree level. A survey of 721 chemistry students by the Higher Education Academy (HEA) found that a quarter of students did not originally expect to extend their mathematical knowledge throughout their degree and 32% found that the mathematical content had been more than they had expected (Shallcross and Yates, 2014). This has a direct impact on both learning and teaching. Staff report significant variability in the mathematical preparedness of new undergraduates, and students report anxiety and uncertainty about the mathematical content of their course (Gadd, 2000; Shallcross and Yates, 2014). Consequently, some universities have implemented introductory mathematics courses in the first year for students who have not taken A-level Mathematics or equivalent (Shallcross et al., 2011; Andersen et al., 2012; Shallcross and Yates, 2014).

2.3 Impact of mathematical backgrounds on performance

The sometimes low mathematics entry requirements for undergraduate chemistry are particularly surprising when considering the strong relationship between the two disciplines. The benchmark statement for undergraduate chemistry requires that students graduate with “numeracy and mathematical skills, including such aspects as error analysis, order-of magnitude estimations, correct use of units and modes of data presentation” (Quality Assurance Agency for Higher Education, 2007, p. 6). However, Shallcross and Yates (2014) argue that this does not place a strong enough emphasis on the mathematical requirements of chemistry. Additionally, other authors argue that a knowledge of algebra, geometry, calculus and modelling has long been considered essential for studying undergraduate chemistry (Bradshaw, 1926; Andraos, 1999).

Perhaps as a consequence of low entry requirements, undergraduate students have been found to struggle with areas of mathematics essential for the study of chemistry (Royal Society of Chemistry, 2008). This has been compounded by the relative lack of mathematics and statistics in A-level Chemistry (Porkess, 2013), though recent changes to the qualification mean that 20% of the question paper must now assess mathematical skills. Whilst difficulties are often focused around basic arithmetic and unit conversion (Gadd, 2000; Scott, 2012), Andersen et al. (2012) also found that students struggled with mechanics, partial derivatives and statistical testing, which are essential for areas such as physical chemistry, thermodynamics and reaction kinetics. Furthermore, two tutors interviewed by ACME (2011) cited a range of areas that their students found difficult:

• Rearranging equations

• Carrying out simple differentiation and integration

• Changing between units

• Plotting graphs

• Recognising trends in data

• Understanding and calculating confidence limits

• Use of logarithms and exponentials

• General problem solving

• ‘Guesstimating’ order of magnitudes

• Aligning the ‘mathematics’ with ‘reality’

• Seeing that the variables in science are the same as the ‘x and y’ in mathematics

(p. 12)

Shallcross et al. (2011) found that students' performance could be dramatically improved by the introduction of an intensive mathematics summer school for new undergraduates without any post-compulsory mathematics qualifications. The summer school covered topics such as basic statistics, integration and differentiation. It was found that the summer school students were the first ever undergraduates without A-level Mathematics to achieve higher than the year average in their first year ‘Theoretical Chemistry’ module. It is interesting to note that the University of Bristol, where the study took place, now requires students to have at least a grade B in A-level Mathematics.

Although very little research has been conducted in the UK that directly compares performance in pre-university mathematics qualifications and undergraduate chemistry, a number of studies in the United States have reported a link between performance in both high-school and college-level mathematics courses and achievement in undergraduate chemistry. Decades ago, Andrews and Andrews (1979) found that students' SAT† Mathematics scores had a positive linear relationship with their first and second semester chemistry grades, and that this was especially strong at the upper end of achievement. More recently, Donovan and Wheland (2009) identified a similar relationship with the ACT‡ Mathematics score, something which may also be used for admission to university. They also noted that students who had higher prior achievement in mathematics were less likely to withdraw from their chemistry course. Additionally, some research has found that students who take university level mathematics or physics courses before studying chemistry modules attain higher grades, particularly in physical chemistry (Derrick and Derrick, 2002; Donovan and Wheland, 2009). This was especially pronounced at the higher end of attainment, with students who attained an A or B grade in undergraduate mathematics courses performing strongly in chemistry courses. Students who only achieved a C grade, however, performed no better than students who did not take any college-level mathematics or physics courses (Derrick and Derrick, 2002; Donovan and Wheland, 2009).

3. The study

3.1 Research aims

This study was conducted in response to reform programme undertaken regarding A-level Mathematics and Further Mathematics (see Section 2.1.3). In order to redesign Further Mathematics and establish what the most useful content would be for students going on to study mathematically demanding subjects at degree-level, this study aimed to establish the view of undergraduates of such subjects who had taken A-level Mathematics regarding their mathematical preparedness.

Previous research in this area has not specifically gathered the views of undergraduates regarding their perceptions of A-level Mathematics and/or Further Mathematics, instead they focussed on topics such as how challenging students find the mathematical component of undergraduate chemistry (e.g.Shallcross and Yates, 2014).

The research outlined in the following sections aimed to answer the following research questions:

(1) Which optional units in A-level Mathematics and/or Further Mathematics did chemistry students find useful as preparation for the mathematical component of their degree?

(2) What motivated chemistry students to take Further Mathematics (if applicable), and what were their experiences of studying it?

(3) Do chemistry students who took A-level Mathematics and/or Further Mathematics believe the qualification(s) were useful preparation for their degree?

(4) Are there any ways in which A-level Mathematics and/or Further Mathematics could be improved to suit the needs of future prospective chemistry students?

Whilst this article focuses on the views of chemistry undergraduates, this work was part of a larger project (Darlington and Bowyer, 2016b) which sought the views of students of a variety of other subjects in the sciences, social sciences and mathematics (see, for example, Darlington and Bowyer, 2016c).

3.2 Method

All 53 UK universities offering single honours undergraduate Bachelor of Science chemistry degrees were contacted via departmental email addresses found on university websites. Chemistry departments were asked to pass on details of an online questionnaire aimed at students who fulfilled two criteria:

(1) Participants must have been in their second year or above (in order that they may reflect on their experiences so far).

(2) Participants must have taken at least AS-level Mathematics, and have taken it since 2006, when the qualification underwent a restructuring.

The survey questioned participants regarding:

• Mathematical background: mathematics A-level qualifications and grades, units studied as part of A-level Mathematics and/or Further Mathematics

• Current studies: degree course, university, year of study

• Perceptions of A-level Mathematics and/or Further Mathematics as preparation for the mathematical component of their degree

• Factors which motivated them to take Further Mathematics (if applicable)

• Experiences of Further Mathematics (if applicable)

The questionnaire was a mixture of closed- and open-ended questions, and multiple choice questions. It was developed by the researchers and an A-level Mathematics expert, before being piloted by three recent graduates of scientific degrees who had taken A-level Mathematics and Further Mathematics.

The research was conducted with ethics committee approval within Cambridge Assessment, which uses the University of Cambridge ethics committee guidelines. Participation in the study was voluntary and anonymous, with informed consent sought before participants answered any of the questionnaire.

4. Results

4.1 Sample

A total of 332 students studying undergraduate degrees in chemistry completed the questionnaire (see Table 2). Unfortunately, it is impossible to calculate a response rate because, though 53 universities were contacted, we cannot be sure how many universities passed on details of the questionnaire to their students, nor can we know how many students were contacted at the universities which took part.
Table 2 Participants' degrees (N = 332)
Degree title Participants
Number %
Chemistry 323 97.3
Chemistry and medicinal chemistry 3 0.9
Applied chemistry 2 0.6
Chemistry and medicinal science 1 0.3
Chemistry and pharmaceutical chemistry 1 0.3
Chemistry and Spanish 1 0.3
Environmental chemistry 1 0.3

The sample was 55% male, which appears to be broadly representative of the overall gender split amongst chemistry undergraduates – in 2009/10 53.3% of students registered on Bachelor's degrees in chemistry were male§ (Institute of Physics, 2012, p. 8).

The participants attended 19 different universities. Over three-quarters (75.6%) of students attended universities ranked in the top 25% of universities for chemistry (Complete University Guide, 2016). Most participants were studying at universities in England (N = 314), with eight students studying in Wales and nine in Scotland.

4.2 Mathematics backgrounds

The data indicate that most participants studied more mathematics than was necessary to meet the entry requirements of their course (see Table 3). 41% of participants stated that they had studied some Further Mathematics. Although students had to have studied at least AS Mathematics to participate in this research, and therefore our sample is biased, participants had generally studied post-compulsory mathematics to a higher level than the overall undergraduate chemistry student body. Vidal Rodeiro and Sutch (2013) found that, in 2010, only 11% of students beginning physical sciences degrees had studied A-level Further Mathematics, with 56.2% having studied A-level Mathematics. This is particularly interesting given the physical sciences categorisation in that study also included materials science, astronomy, geology, oceanography, geography, environmental science and physics, a degree subject for which Further Mathematics is often a required or recommended A-level subject for admission.
Table 3 Highest mathematical A-level qualifications (N = 332)
Qualification(s) Frequency Proportion (%)
AS-level Mathematics only 7 2.1
A-level Mathematics only 190 57.2
A-level Mathematics + AS-level Further Mathematics 46 13.9
A-level Mathematics + A-level Further Mathematics 89 26.8

The data in Table 4 indicate that participants were high-achievers in both Mathematics and Further Mathematics, compared to both all A-level candidates in 2016 and all undergraduates of the physical sciences in 2011 (the most up-to-date data available). This indicates that the sample is skewed towards the higher end of A-level achievement.

Table 4 Participants' AS- and A-level Mathematics and/or Further Mathematics grades
AS- or A-level grade % Students
Mathematics Further Mathematics
Participants (n = 325) All candidates (2016) All physical sciences undergraduates (2011) Participants (n = 78) All candidates (2016)
Additional data obtained from the Joint Council for Qualifications (2016) and Vidal Rodeiro (2012).
A*/A 87.6 38.7 57.8 84.8 54.7
B 8.1 19.7 23.2 8.1 18.1
C 0.6 14.9 12.1 4.0 11.5
D 1.5 10.5 4.9 2.4 7.1
E 1.5 7.1 2.0 0.0 4.5
Ungraded (fail) 0.0 8.2 0.0 0.0 4.1

Similar proportions of participants studied Mechanics or Statistics units during their A-level studies, with a smaller number having studied any Decision Mathematics. It was rare for participants to have studied more than two units in the same applied strand (see Fig. 3), although half of the students who had taken Mechanics had studied at least two mechanics units. Female students were significantly more likely than male students to have studied more than one Statistics unit (Fisher's exact test statistic = 7.499, p = 0.040).


image file: c6rp00170j-f3.tif
Fig. 3 Applied A-level units taken.

4.3 Experiences of optional units

Participants were asked whether they considered non-compulsory A-level units to be ‘very useful’, ‘somewhat useful’ or ‘not useful’ as preparation for the mathematical content of their degree (see Fig. 4).
image file: c6rp00170j-f4.tif
Fig. 4 Students' views of the usefulness of optional units.

Further Pure Mathematics units (only available through the study of Further Mathematics) were considered to be the most useful units, with 61.2% of participants who had taken them reporting that they were very useful preparation for the mathematical demands of their chemistry degree. However, opinions about the other units were considerably more negative. Whilst Mechanics was considered to be more useful than either Statistics or Decision Mathematics, with 82.8% stating that Mechanics units were at least somewhat useful, only 26.1% of students reported that they found them very useful as preparation for their degree.

Statistics units were reported to be very or somewhat useful preparation by 65.8% of students, with 30.6% reporting that they were not useful. Opinions of Decision Mathematics units were generally negative, with a sizeable majority (84.1%) stating that they were not useful as preparation for the mathematical content of their undergraduate course.

4.4 Motivations for studying Further Mathematics

The 135 students who had studied either AS- or A-level Further Mathematics were asked whether certain factors had influenced their decisions to study Further Mathematics. The statements used in this question were taken from a study by the Qualifications and Curriculum Authority (2007) regarding factors which motivated students to take A-level Mathematics. The strongest determinants on students' decisions appear to be prior enjoyment of, and attainment in, school mathematics (see Fig. 5).
image file: c6rp00170j-f5.tif
Fig. 5 Students' motivations for studying Further Mathematics (n = 133).

The four factors that participants most frequently reported as having a strong influence were:

Enjoyment of school mathematics. Of all of the factors suggested, ‘I enjoyed school maths’ had the highest proportion of participants stating that this influenced them a lot in their decision making. Only 6.8% reported that this did not influence their decision.
Prior attainment. Nearly 75% of participants claimed that coping well with GCSE Mathematics influenced them a lot. 88.6% reported that they were influenced to some extent by their perception that they were better at mathematics than at other subjects.
Synergy with other A-level subjects. The factor which had the third-greatest proportion of participants claiming that it influenced them a lot was a belief that Further Mathematics fitted well with their other subject choices. Only five students reported this did not influence them, with 72.9% reporting that it influenced them a lot.
Utility of Further Mathematics. 72.5% of participants reported that they were influenced a lot by a belief that it would be a useful qualification to have.

Gender differences were found in students’ responses to some of these questions. Women were found to be significantly more likely than men to have been influenced to study Further Mathematics because they believed that they were better at mathematics than other subjects (Fisher's exact test statistic = 7.672, p = 0.024). Men were significantly more likely than women to have been influenced by a belief that the topics in Further Mathematics looked interesting (χ2(2) = 13.831, p = 0.001).

4.5 Experiences of Further Mathematics

Participants were also asked whether they agreed about a series of statements about their experiences of studying Further Mathematics. The data indicate that, broadly, participants' experiences were positive (see Table 5).
Table 5 Students' experiences of Further Mathematics
Number of participants (%)
Strongly agree Agree Neither agree nor disagree Disagree Strongly disagree Unsure
I'm glad I did Further Maths 74 (55.2%) 49 (36.6%) 8 (6.0%) 2 (1.5%) 1 (0.7%) 0 (0.0%)
In my first year at university, we were taught material that I had learned in Further Maths 73 (54.5%) 50 (37.3%) 4 (3.0%) 6 (4.5%) 0 (0.0%) 1 (0.7%)
I enjoyed Further Maths 54 (40.3%) 52 (38.8%) 16 (11.9%) 10 (7.5%) 2 (1.5%) 0 (0.0%)
I found Further Maths challenging 47 (35.1%) 72 (53.7%) 8 (6.0%) 4 (3.0%) 2 (1.5%) 1 (0.7%)
Further Maths was my most difficult A-level 41 (30.6%) 25 (18.7%) 17 (12.7%) 37 (27.6%) 11 (8.2%) 3 (2.2%)
Most people on my university course studied Further Maths 11 (8.2%) 25 (18.7%) 27 (20.1%) 44 (32.8%) 21 (15.7%) 6 (4.5%)

Despite the apparent difficulty of Further Mathematics, with 89.4% of participants agreeing or strongly agreeing that they had found it challenging, 79.1% agreed or strongly agreed that they had enjoyed the subject. Additionally, 91.8% strongly agreed or agreed that they were glad they had studied it. The high percentage of participants stating that they were glad they had studied it possibly reflects the strong link between Further Mathematics and undergraduate chemistry content. 92.5% reported that they were taught material from Further Mathematics in their first year of university. This finding is especially significant when considering the low numbers of students beginning degrees in the physical sciences, which includes chemistry, having studied Further Mathematics.

Additionally, participants who had studied only the first Further Pure Mathematics unit were less likely than participants who had studied at least two Further Pure Mathematics units to strongly agree that they were glad they had studied Further Mathematics (Fisher's exact test statistic = 47.817, p = 0.023). This suggests that studying multiple Further Pure Mathematics units may be particularly beneficial, presumably because of the greater exposure to advanced calculus content.

4.6 A-level as preparation for the mathematical demands of undergraduate chemistry

Participants were also asked whether, overall, A-level Mathematics and/or Further Mathematics had been good preparation for the mathematical components of their degree. The responses were highly positive, with no participant reporting that Further Mathematics had been bad preparation (see Fig. 6). Specifically, 82.5% of participants who had taken AS- or A-level Mathematics and 86.4% of participants who had taken some Further Mathematics described the qualification as good preparation for undergraduate chemistry.
image file: c6rp00170j-f6.tif
Fig. 6 The A-levels as preparation for undergraduate chemistry.

Whilst perceptions of both A-levels were broadly good, perceptions of Further Mathematics were found to be associated with the number of Mechanics units a participant had studied. Participants who had studied more than one Mechanics unit were more likely to report that Further Mathematics was good preparation (Fisher's exact test statistic = 10.173, p = 0.026). This is particularly interesting, as a similar relationship was not found with the number of Further Pure Mathematics units studied. Additionally, the number of Mechanics units studied did not seem to impact on how useful students perceived Mechanics units to be.

4.7 Improvements to A-level Mathematics and Further Mathematics

Participants were also asked two open response questions: whether there were any topics they would have found useful to have been included in A-level Mathematics and Further Mathematics, and whether there were any improvements that could be made to these A-levels in order to better prepare students for the mathematical components of chemistry degrees.

There were 219 responses regarding additional topics. The most commonly cited topic was matrix algebra, followed by more complex differentiation and integration. However, some students acknowledged that they would have encountered more differentiation if they had studied Further Mathematics, indicating that this is a useful A-level for potential undergraduate chemists to study. The most common suggestions are outlined in Table 6.

Table 6 Chemistry participants' suggested topics for inclusion in A-level
Topic Current A-level Reformed A-level
Mathematics Further Mathematics Mathematicsa Further Mathematicsb
a Department for Education, 2016a. b Department for Education, 2016b.
Matrices Matrix algebra

Eigenvalues and eigenvectors

 ✓

✓

 ✓

✓
Differentiation Partial differentiation

Second order differential equations

 ✓ ✓ ✓ ✓
Integration Partial integration

Multiple integrals

 ✓ ✓ ✓ ✓
Complex numbers ✓ ✓
Imaginary numbers ✓ ✓
Vector calculus
Mechanics Quantum mechanics Optional Optional ✓ Optional
Probability and statistics Error propagation

Statistical testing

 Optional Optional ✓ Optional
Fourier series
Taylor series

Table 6 shows that some participants had suggested topics which were already (or will be) available as part of A-level Mathematics and/or Further Mathematics. This is a consequence of the optionality afforded to students in their unit choices.

199 participants gave suggestions for improvements to A-level Mathematics or Further Mathematics. The most frequent suggestion for improvement was making the mathematics covered in both A-levels more relevant to chemistry, by introducing questions that were applied in chemical contexts. A minority of these participants felt that this would make mathematics seem less abstract and promote better understanding of the material. It was suggested that this could be helpful for other sciences such as physics and engineering, consequently proving better preparation for university study in a range of subjects. Further suggestions indicated that the inclusion of more problem-solving questions would be welcome.

The majority of participants who specifically referred to depth suggested that the A-levels could be improved by going into greater depth, particularly in integration and differentiation, where a smaller proportion of responses repeated calls for more advanced calculus to be covered in A-level Mathematics.

Assessment was a common concern, with the majority of participants commenting on examination at A-level reporting that they had found the exams repetitive and predictable. The most commonly suggested improvement was setting questions in an applied context, especially with Core Pure Mathematics or Further Pure Mathematics units. Additionally, most participants who specifically commented on difficulty indicated that an increase in difficulty at A-level would be beneficial as preparation for undergraduate study.

Despite the above concerns and suggestions, many participants reported that they were happy with both the difficulty and content of A-level Mathematics and Further Mathematics, which they believed had served them well in the transition to university study. In particular, most of those who had taken Further Mathematics noted that it had been good preparation for the mathematical demands of their degree (supporting responses to an earlier question – see Fig. 5). Furthermore, other comments about Further Mathematics, although less common, suggested that Further Mathematics should either be compulsory for admission to their course, or that some topics could be moved from Further Pure Mathematics units into A-level Mathematics, predominantly matrices, complex numbers and further differentiation and integration.

5. Limitations

Due to the fact that this study was self-selecting, there are a number of limitations associated with how the data can be interpreted.

Participation was not only self-selecting in terms of students choosing to respond to the questionnaire, but it also required self-selection on the part of university chemistry departments in their decision to forward details of the study to their students or not. This is why national data has been given in Table 4 so as to give an insight into the representativeness of the sample. It shows that the participants in this study belong to the higher-attaining end of the cohort, even when discounting the fact that A-level Mathematics is not studied by all prospective undergraduate chemists. Additionally, it could be that students who felt particularly strongly about their mathematical preparedness and its impact on their transition to undergraduate study (either positively or negatively) felt more compelled to take part than those with more neutral opinions.

This study only incorporates the views of students who had taken A-level Mathematics and/or Further Mathematics. We cannot contrast their responses with students who did not take these A-levels, or with students who took alternative qualifications. However, the Shallcross et al. (2011) study incorporated the views of students who had not taken A-level Mathematics in its investigation. It is possible that students who had taken qualifications in other countries had different perceptions of their mathematical preparedness.

Finally, participants in this study were current chemistry students. We are unable to know how well the students will perform in their degrees overall, and therefore are unable to make any claims regarding whether taking A-level Mathematics and/or Further Mathematics results in better attainment or degree outcomes. This is something which would be a useful subject of further research.

6. Discussion and conclusion

The findings of this study show that undergraduate chemistry students who have studied A-level Mathematics or Further Mathematics consider them to be good preparation for the mathematical content of their degree. In particular, the support for Further Mathematics was found to be very high, with no participant reporting that it had been bad preparation. This is especially significant, as few universities require students to have studied A-level Mathematics for admission to undergraduate chemistry, and none require Further Mathematics. Further Pure Mathematics units were also considered to be the most useful in terms of preparation for university study. Of the applied units at A-level, Mechanics units were reported to be the most useful, although opinions were somewhat more lukewarm than for Further Pure Mathematics units. Statistics and especially Decision Mathematics units appear to be less useful for undergraduate chemistry, indicating that potential chemists would benefit from specialising in Mechanics during their A-level studies. However, this is something which will only be possible through the study of Further Mathematics after qualifications are reformed in September 2017.

The positive opinions about Further Mathematics are perhaps unsurprising when the type of mathematics involved in the study of chemistry is considered. The Royal Society of Chemistry (2009) indicate that integration, differentiation and functions are all essential skills for undergraduate chemists, and these topics are all studied in much more depth in Further Mathematics. The benefit of further study in this area is reflected in the positive opinions participants expressed about Further Pure Mathematics units, with 94% of reporting that they had been very or somewhat useful as preparation for their degree.

Additionally, participants' responses indicate that, whilst universities may be reluctant to require or explicitly recommend Further Mathematics to potential students, new students encounter material covered in Further Pure Mathematics units at undergraduate level. Of the participants who had studied Further Mathematics, 92.5% reported that they had been taught material that they had covered in the A-level during their first year of university study. This indicates that there is a strong relationship between the mathematics studied in undergraduate chemistry and the material covered in Further Mathematics. It also suggests that universities are pitching their teaching towards the minimum entry requirements that they are asking of applicants. That is, by learning material in undergraduate chemistry that is part of Further Mathematics, some students can have an advantage through already being familiar with the material. This is particularly the case in the transition from secondary to tertiary chemistry when students will not only be adapting to new ways of learning and studying, and living away from home, but also to new and more challenging topics. The fact that most students reported that they were glad they had taken A-level Further Mathematics suggests that, whilst it might not be necessary for them to have taken it to adapt to undergraduate chemistry, it certainly was beneficial.

Most participants suggested that more advanced differentiation and integration, matrix algebra and complex numbers should be incorporated in A-level Mathematics. However, the proposed subject content for the reformed A-levels indicates that matrices and advanced calculus will remain in Further Mathematics, although the new prescribed structure for Mathematics means that all students will have to study some mechanics content (Department for Education, 2014a). The participants in this study who did take Further Mathematics appear to have been at a significant advantage when compared to peers who only studied AS- or A-level Mathematics, due to their greater familiarity with and knowledge of calculus. Given widespread concern in the higher education community about the number of students studying undergraduate chemistry, it is unlikely that universities will begin to require Further Mathematics, fearing that more stringent entry requirements could deter potential students. Nevertheless, the findings of this study indicate that universities would benefit from students having some exposure to the type of content covered in Further Mathematics prior to beginning their degree and therefore it could be beneficial to recommend its study.

In conclusion, concern in chemistry departments that the introduction of post-compulsory mathematics requirements would dissuade some prospective undergraduates belies the large proportion of students in this study who found both A-level Mathematics and Further Mathematics to be good preparation for their degree. Additionally, the large number of departments which offer additional mathematics support for undergraduate chemists (Shallcross and Yates, 2014) highlights a need for students to be better prepared for the mathematical demands of tertiary chemistry. Concerns about widening participation are certainly valid, especially considering that not all schools are able to teach Further Mathematics and the number of chemistry undergraduates is already considered to be vulnerable at a time when A-level Mathematics is rarely an entry requirement. However, it appears that students considering undergraduate chemistry would benefit from studying at least A-level Mathematics, and university departments might consider at least recommending the study of AS- or A-level Further Mathematics to prospective undergraduates.

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Footnotes

† The ‘Scholastic Assessment Test’ is widely used for university admissions in the United States of America, and include mathematics and critical reading sections.
‡ The ‘American College Testing’ examination is used for American university admissions and comprises four tests: mathematics, English, science and reading.
§ This is the most recent data available.
This journal is © The Royal Society of Chemistry 2016